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Fibonacci Numbers

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... They show up in nature, they show up in math, and they've got some beautiful properties.

Challenge Quizzes

Level 3

         

Compute \[ \frac{1}{2} + \frac{1}{4} + \frac{2}{8} + \frac{3}{16} + \frac{5}{32} + \dots + \frac{F_k}{2^k} + \dots \] where \( F_k\) represents the the \( k^\text{th} \) term of the Fibonacci sequence: \(1, 1, 2, 3, 5, 8, 13, ... \)

Find the last digit of the 123456789-th Fibonacci number.

In the Fibonacci sequence, \(F_{0}=1\), \({F_1}=1\), and for all \(N>1\), \(F_N=F_{N-1}+F_{N-2}\).

How many of the first 2014 Fibonacci terms end in 0?

The Fibonacci sequence is defined by \(F_1 = 1, F_2 = 1\) and \( F_{n+2} = F_{n+1} + F_{n}\) for \( n \geq 1 \).

Find the greatest common divisor of \(f_{484}\) and \(f_{2013}\).

\[\sum_{n=0}^{\infty} \frac{F_{n}}{3^{n}}= \ ? \]

Details and Assumptions

\(F_{n}\) is the \(n^\text{th} \) Fibonacci number, with \(F_{1} = F_{2} = 1\).

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